Approximation properties of Cesàro means of Vilenkin-Fourier series

This PhD thesis focuses on the investigation of approximation properties of Cesàro means of the Vilenkin-Fourier series. In particular, we obtain some new inequalities related to the rate of Lp approximation by Cesàro means of the Vilenkin-Fourier series of functions from Lp. These inequalities imply sufficient conditions for the convergence of Cesàro means of the Vilenkin-Fourier series in the Lp−metric in terms of the modulus of continuity. Furthermore, we also proved the sharpness of these conditions. In particular, we find a continuous function under some condition of the modulus of continuity, for which Cesàro means of the Vilenkin-Fourier series diverge in the Lp− metric. This PhD thesis consists of three main Chapters, based on five papers. At first, we have an Introduction, where we give a general overview of fundamental definitions and notations, followed by historical and new results, on which our study is based and inspired. We also give a formulation of our main results in this general frame and review some auxiliary results, that are significant for the proofs of our new theorems in the next main chapters. In Chapter 1, we investigate the approximation properties of Cesàro means of negative order of the one-dimensional Vilenkin-Fourier Series. In particular, we derive sufficient conditions for the convergence of the means σ−α n (f, x) to f(x) in the Lp− metric in terms of the modulus of continuity. Moreover, we prove the sharpness of these conditions. Chapter 2 is focused on a new approach to investigate the rate of Lp approximation by Cesàro means of negative order of the two-dimensional Vilenkin-Fourier Series of functions from Lp. In particular, we derived a necessary and sufficient condition for the convergence of Cesàro (C,−α,−β) means with α,β ∊ (0, 1) in terms of the modulus of continuity. Some corresponding sharpness results are proved also in this case. Chapter 3 is devoted to deriving some new results concerning the behavior of Cesàro (C,−α) means of the quadratic partial sums of double Vilenkin-Fourier series. The new results are sharp also in this case

A study of bounded operators on martingale Hardy spaces

The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the "State of the art", but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some T means, which are "inverse" summability methods of Nörlund, but only in the case when their coefficients are monotone. The main body of the PhD thesis consists of seven papers (Papers A -- G). We now continue by describing the main content of each of the papers. In Paper A we investigate some new strong convergence theorems for partial sums with respect to Vilenkin system. In Paper B we characterize subsequences of Fejér means with respect to Vilenkin systems, which are bounded from the Hardy space Hp to the Lebesgue space Lp for all 0 < p < ½. We also proved that this result is in a sense sharp. In Paper C we find necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space Hp to the Lebesgue space Lp for all 0 < p < ½. In Paper D we prove and discuss some new (Hp, weak-Lp) type inequalities of maximal operators of T means with respect to Vilenkin systems with monotone coefficients. We also apply these results to prove a.e. convergence of such T means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out. In Paper E we prove and discuss some new (Hp, Lp) type inequalities of weighted maximal operators of T means with respect to the Vilenkin systems with monotone coefficients. We also show that these inequalities are the best possible in a special sense. Moreover, we apply these inequalities to prove strong convergence theorems of such T means. We also show that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out. In Paper F we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series. In Paper G we investigate (Hp, Lp) type inequalities for weighted maximal operators of Nörlund logaritmic means, for 0 < p < 1. Moreover, we apply these inequalities to prove strong convergence theorems of such Nörlund logaritmic means. These new results are put into a more general frame in an Introduction, where, in particular, a comparison with some new international research and broad view of such interplay between applied mathematics and engineering problems is presented and discussed